Effective behavior of random media: an error analysis

CTMS Adventures In Theory Lectures Seminar

Felix Otto (Max Planck Institute Leipzig)

Tuesday, May 9, 2017 -
3:15pm to 4:15pm
130 Physics

Heterogeneous media, like a sediment, are often naturally described in statistical terms. How to extract their effective behavior on large scales, like the permeability in Darcy's law, from the statistical specifications? A practitioners numerical approach is to sample the medium according to these specifications and to determine the permeability in the Cartesian directions by imposing simple boundary conditions. What is the error made in terms of the size of this ``representative volume element''?

Our interest in what in applied mathematics is called ``stochastic homogenization'' and in stochastic analysis ``quenched invariance principles'' grew out from developing tools to answer this questions rigorously.

Two further quantitative questions I'd like to address in this talk are: In which sense is there an effective multipole expansion in a random medium, can its coefficients be predicted without knowing the realization of the medium away from the localized source? To which extent can fluctuations of macroscopic observables be characterized?